Indian Institute of Technology Kharagpur (IIT-K) 2010 M.Sc Mathematics & Computing Matrix Algebra - Question Paper
Indian Institute of Technology, Kharagpur
Deparment:EC,MI and IE.
MA20107, Matrix Algebra
End semester(Autumn) 2010 No. of Students: 100
Time:3 hrs Full Marks: 100
Indian Institute of Technology Kharagpur Department: EC, MI and IE.
MA20107 Matrix Algebra Autumn End Semester Examination, 2010 No. of Students: 100
Full Marks: 50, Time: 3 Hrs.
INSTRUCTION: Answer any 10 questions. Each question carries equal marks.
1. (a) If Ui,U2, Uz are subspaces of a finite dimensional vector space, then
dim({/i + U% + Uz) ~ dimUi + dimt/2 +dim6r3 dim fl t/2) - dim(6ri D U$) -dim({72 fl Uz) + dim({/i V\U%C\Uz). Prove this or give a counter example.
(b) If A is an eigen value of a real orthogonal matrix A, prove that j is also an eigen value of A.
(2+3 5 marks)
2. (a) Let V be a vector space over R and S C V a subset (not necessarily a subspace).
Show that the following two conditions on S are equivalent:
(i) S is non-empty; and if x, y S and AeR, then Ax + (1 A)y 6 S.
(ii) There is a vector v V and a subspace W of V, such that x S o x v
W. (Such an S is called an affine subspace of V).
(b) If S,T are subsets of V(F), then show that S C L(T) => L(S) C L(T), where L(S),L(T) denote the linear span of S and T respectively.
(4+1 --- 5 marks)
3. (a) Let V and W be finite dimensional vector spaces of the same dimension over a
field F and T : V ~ W be a linear mapping. Then show that T is one-to-one T is onto.
(b) If a, 0 are vectors in an inner product space V (F) and a,b G F, then prove that
(i) ||aa + 6/5||2 = [a|2|ja||2 + ab(a,fi) + ab(0,a) + |i>|2||||2-
(ii) Re(a,/J) = i||o: + )3|p -
(2+3 5 marks)
P. T. O
4. (a) Let T be the linear operator on M3 defined by
T(xilx2,x3) (x! - x2yx2 ~ xXtxi - x3).
Find the matrix representation of T with respect to the ordered basis {cti, 0$, <*3)1 where <*1 = (1,0, l),a2 = (0,1,1), a3 = (1,1,0).
(-l)n+r
(b) Determine whether ln(/+/l) =
n=0
!\ 2 1 A= 0 3 5 \0 -1 3
An is well-defined for the matrix
n
(3+2 = 5 marks)
5. (a) If A be an eigen value of a matrix A of order n x n and W\ be the set of
eigen vectors corresponding to eigen value A together with {0}, that is, W\ = {X\AX = XX], then show that the set W\ is a subspace of Vn(F).
(b) Let M = , where A and B are square matrices. Show that the minimal
polynomial m(i) of M is the least common multiple of the minimal polynomials g(t) and h(t) of A and B respectively.
(2+3 = 5 marks)
6. (a) Prove that the minimal polynomial of a matrix is a divisor of every polynomial
that annihilates the matrix.
(b) Examine whether the functions /(f) = sin t, g(t) = cos i, h(t) = t are linearly independent.
(3+2 = 5 marks)
7. (a) Find two linear operators T and S on (R) such that TS = O but ST O.
/ 3 2 4 \
(b) Find the expression A24 3-A15 if A I 0 1 0 j.
\-l -3 ~\)
(2+3 = 5 marks) P. T. O
8. (a) Find a generalized eigen vector of type 3 corresponding to the eigen value A = 7
?7 1 2n
for the matrix A = | 0 7 1 VO 0 7,
(b) Use Gram-Schmidt process to obtain an orthonormal basis of the subspace of the Euclidean space R4 with the standard inner product, generated by the linearly independent set {(1,1,0,1), (1,1,0,0), (0,1,0,1)}.
(2+3 = 5 marks)
9. (a) Assume that X = 2 is the only eigen value for a 5 x 5 matrix A. Find a matrix in
Jordan canonical form similar to A if the complete set of pk numbers associated
(b) Determine a canonical basis for A =
|
5 = P4 |
P3 |
= |
P2 ~ | |
|
( 4 |
1 |
1 |
2 |
2\ |
|
-1 |
2 |
1 |
3 |
0 |
|
0 |
0 |
3 |
0 |
0 |
|
0 |
0 |
0 |
2 |
1 |
|
\0 |
0 |
0 |
1 |
V |
(1+4 = 5 marks)
10. (a) Let T be a linear transformation from a vector space U into a vector space V
with kerT {6}. Show that there exist vectors c*i and q2 in U such that c*! a2 and Ta% = Taa.
(b) Prove that two vectors a and /? in a real inner product space V are orthogonal if and only if ||o; + /3||2 = |MI2 + ||/?||2- Does the result hold if V is a complex inner product space? Give reasons for your answer.
(2+3 = 5 marks)
11. (a) Prove that a chain is a linearly independent set of vectors.
(b) Prove Parsevals theorem which states that if {ft, ft,..., (3nj be an orthonormal basis of a Euclidean space V, then for any vector oc in V, we have | jo:] ]2 = ci + 2 + + cni where c* is the scalar component of a along ft, i = 1,2,..., n.
(3+2 = 5 marks)
3
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Attachment: |
| Earning: Approval pending. |
